# On the Picard number of Fano 6-folds with a non-small contraction

## TL;DR

This paper proves a conjecture relating the Picard number and pseudo-index of Fano 6-folds, confirming the inequality in specific cases where the variety admits certain types of contractions.

## Contribution

It establishes the validity of Mukai's conjecture for Fano 6-folds under conditions of fiber type or absence of small contractions.

## Key findings

- The conjecture holds for Fano 6-folds with a fiber type contraction.
- The conjecture holds for Fano 6-folds with no small contractions.
- Provides a classification framework for these Fano varieties.

## Abstract

A generalization of S. Mukai's conjecture says that if $X$ is a Fano $n$-fold with Picard number $\rho_X$ and pseudo-index $i_X$, then $\rho_X(i_X-1) \leq n$, with equality if and only if $X \cong (\mathbb{P}^{i_X-1})^{\rho_X}$. In this paper, we prove that this conjecture holds if $n=6$ and either $X$ admits a contraction of fiber type or $X$ admits no small contractions.